Download Lines of Force

JAMES CLERK MAXWELL
On Faraday's Lines of Force*
James Clerk Maxwell (1831-1879), a Scottish physicist, first professor of experimental physics at
Cambridge University, is known today chiefly for his studies of electromagnetism, epitomized in
"Maxwell's equations." Basing his work on the concepts and experiments of Faraday, Maxwell conceived
a mathematical theory encompassing the related phenomena of electricity and magnetism, in which he
predicted the possible existence of electromagnetic waves, the basis of transmission of radio, radar, and
television. Above all, this theory had a unifying influence on physical thought by showing that light itself
could be considered as an electromagnetic effect. In 1859 and 1860 Maxwell read a two-part paper
entitled "Illustrations of the Dynamical Theory of Gases" to the British Association. In many ways as
revolutionary for science as the theory of evolution, Maxwell's marked the first full-scale presentation of
a physical theory in which the basic laws were shown to have a statistical rather than a causal basis. The
following selection, taken from Maxwell's first electrical paper, was read to the Cambridge Philosophical
Society on December 10, 1855 and on February 11, 1856, just a year after the author, aged twenty-three,
had received his Bachelor of Arts degrees The present selection includes the whole of the introductory
part of the paper. the remainder being unintelligible to those unskilled in the language of mathematical
abstraction.
THE present state of electrical science seems peculiarly unfavourable to speculation. The
laws of the distribution of electricity on the surface of conductors have been analytically
deduced from experiment; some parts of the mathematical [328] theory of magnetism 1 are
established, while in other parts the experimental data are wanting; the theory of the conduction
of galvanism and that of the mutual attraction of conductors have been reduced to mathematical
formulae, but have not fallen into relation with the other parts of the science. No electrical
theory can now be put forth, unless it shews the connexion not only between electricity at rest
and current electricity, but between the attractions and inductive effects of electricity in both
states. 2 Such a theory must accurately satisfy these laws, the mathematical form of which is
known, and must afford the means of calculating the effects in the limiting cases where the
known formulae are inapplicable. In order therefore to appreciate the requirements of the
science, the student must make himself familiar with a considerable body of most intricate
mathematics, the mere retention of which in the memory materially interferes with further
progress. The first process therefore in the effectual study of the science, must be one of
simplification and reduction of the results of previous investigation to a form in which the mind
can grasp them. The results of this simplification may take the form of a purely mathematical
formula or of a physical hypothesis. In the first case we entirely lose sight of the phenomena to
be explained; and though we may trace out the consequences of given laws, we can never obtain
more extended views of the connexions of [329] the subject. If, on the other hand, we adopt a
physical hypothesis, we see the phenomena only through a medium, and are ,liable to that

From Transactions of the Cambridge Philosophical Society (1856). Reprinted from Science Before Darwin:
An anthology of British Scientific Writing in the Early Nineteenth Century, ed. I. Bernard Cohen and Howard Mumford
Jones (London: Andre Deutsch, 1963), 327-336.
1
In Maxwell's day the word galvanism was used to denote electricity evolved by chemical action, as in the case of a battery.
In the case of "electricity at rest," two insulated charged bodies with opposite charges will attract each other, just as
in "current electricity" parallel wires bearing electric currents in the same direction will attract each other. By "inductiv e
effect," Maxwell referred to the fact that a body with an electric charge will, by being in the neighborhood of a
conducting body, produce the effects of charge in it.
2
blindness to facts and rashness in assumption which a partial explanation encourages. We must
therefore discover some method of investigation which allows the mind at every step to lay hold
of a clear physical conception, without being committed to any theory founded on the physical
science from which that conception is borrowed, so that it is neither drawn aside from the subject
in pursuit of analytical subtleties, nor carried beyond the truth by a favourite hypothesis.
In order to obtain physical ideas without adopting a physical theory we must make ourselves
familiar with the existence of physical analogies. By a physical analogy I mean that partial similarity
between the laws of one science and those of another which makes each of them illustrate the
other. Thus all the mathematical sciences are founded on relations between physical laws and laws
of numbers, so that the aim of exact science is to reduce the problems of nature to the
determination of quantities by operations with numbers. Passing from the most universal of all
analogies to a very partial one, we find the same resemblance in mathematical form between two
different phenomena giving rise to a physical theory of light.
The changes of direction which light undergoes in passing from one medium to another,
are identical with the deviations of the path of a particle in moving through a narrow space in
which intense forces act. This analogy, which extends only to the direction, and not to the
velocity of motion, was long believed to be the true explanation of the refraction of light; 3 [330]
and we still find it useful in the solution of certain problems, in which we employ it without danger,
as an artificial method. The other analogy, between light and the vibrations of an elastic medium,
extends much farther, but, though its importance and fruitfulness cannot be over-estimated, we must
recollect that it is founded only on a resemblance in form between the laws of light and those of
vibrations. By stripping it of its physical dress and reducing it to a theory of "transverse
alternations,"44 we might obtain a system of truth strictly founded on observation, but probably
deficient both in the vividness of its conceptions and the fertility of its method. I have said thus
much on the disputed questions of Optics, as a preparation for the discussion of the almost
universally admitted theory of attraction at a distance.
We have all acquired the mathematical conception of these attractions. We can reason about them
and determine their appropriate forms or formulae. These formulæ have a distinct mathematical
significance, and their results are found to be in accordance with natural phenomena. There is no
formula in applied mathematics more consistent with nature than the formula of attractions, and no
theory better established in the minds of men than that of the action of bodies on one another [331]
at a distance. The laws of the conduction of heat in uniform media appear at first sight among the most
different in their physical relations from those relating to attractions. The quantities which enter into them are
temperature, flow of heat, conductivity. The word force is foreign to the subject. Yet we find that the mathematical
laws of the uniform motion of heat in homogeneous media are, identical in form with those of attractions
varying inversely as the square of the distance. We have only to substitute source of heat, for centre of attraction, flow
3
According to Newton's theory of light, in which optical phenomena were thought to be due to the motion of a series of
particles, it was held that when light goes from air into glass or water it must suffer an increase in speed, as a
material particle would. In fact, however, unlike the stream of particles in the circumstances described by
Maxwell, tight suffers a diminution of speed when it is refracted and enters water or glass from air — a fact only
recently discovered in Maxwell's day.
4
In Maxwell's day, the theory that light consists of a series of vibrations of some kind of an elastic or
ethereal medium had been generally accepted. By "transverse," it was meant that the vibrations were at right
angles to the direction of motion, much as water waves cause when a stone is dropped on the surface and the
waves spread out in a direction at right angles to the motion of the stone.
of heat for accelerating effect of attraction at any point, and temperature for potential, and the solution of a problem in
attractions is transformed into that of a problem in heat.5
Now the conduction of heat is supposed to proceed by an action between contiguous parts of a medium,
while the force of attraction is a relation between distant bodies, and yet, if we knew nothing more than is
expressed in the mathematical formulae, there would be nothing to distinguish between the one set of
phenomena and the other.
It is true, that if we introduce other considerations and observe additional facts, the two subjects will assume
very different aspects, but the mathematical resemblance of some of their laws will remain, and may still be made
useful in exciting appropriate mathematical ideas.
It is by the use of analogies of this kind that I have attempted to bring before the mind, in a convenient and
manageable form, those mathematical ideas which are necessary to the study of the phenomena of electricity. The
methods are generally those suggested by the processes of reasoning which are found in the [332] researches of
Faraday, and which, though they have been interpreted mathematically by Prof. Thomson and others, arc
very generally supposed to be of an indefinite and unmathematical character, when compared with those
employed by the professed mathematicians. By the method which I adopt, I hope to render it evident
that I am not attempting to establish any physical theory of a science in which- I have hardly made a
single experiment, and that the limit of my design is to shew how, by a strict application of the ideas and
methods of Faraday, the connexion of the very different orders of phenomena which he has discovered
may be clearly placed before the mathematical mind. I shall therefore avoid as much as I can the
introduction of anything which does not serve as a direct illustration of Faraday's methods, or of the
mathematical deductions which may be made from them. In treating the simpler parts of the subject I
shall use Faraday's mathematical methods as well as his ideas. When the complexity of the subject
requires it, I shall use analytical notation, still confining myself to the development of ideas originated by
the same philosopher.
I have in the first place to explain and illustrate the idea of "lines of force."
When a body is electrified in any manner, a small body charged with positive electricity, and
placed in any given position, will experience a force urging it in a certain direction. If the small body,
be now negatively electrified, it will be urged by an equal force in a direction exactly opposite.
The same relations hold between a magnetic body and the north or south poles of a small
magnet. If the north pole is urged in one direction, the south pole is urged in the opposite direction.
[333] In this way we might find a line passing through any point of space, such that it represents the
direction of the force acting on a positively electrified particle, or on an elementary north pole, and
the reverse direction of the force on a negatively electrified particle or an elementary south pole.
Since at every point of space such a direction may be found, if we commence at any point and draw
a line so that, as we go along it, its direction at any point shall always coincide with that of the
resultant force at that point, this curve will indicate the direction of that force for every point
through which it passes, and might be called on that account a line of force. We might in the same way
draw other lines of force, till we had filled all space with curves indicating by their direction that of
the force at any assigned point.
We should thus obtain a geometrical model of the physical phenomena, which would tell us the
direction of the force, but we should still require some method of indicating the intensity of the force at
5
Maxwell at this point noted that the analogy between heat and attraction had first been pointed out by Professor
William Thomson, later Lord Kelvin.
any point. If we consider these curves not as mere lines, but as fine tubes of variable section carrying
an incompressible fluid, then, since the velocity of the fluid is inversely as the section of the tube, we
may make the velocity vary according to any given law, by regulating the section of the tube, and in
this way we might represent the intensity of the force as well as its direction by the motion of the
fluid in these tubes. This method of representing the intensity of a force by the velocity of an
imaginary fluid in a tube is applicable to any conceivable system of forces, but it is capable of great
simplification in the case in which the forces are such as can be explained by the hypothesis of
attractions varying inversely as the square of the distance, such as those observed in electrical [ 334 ]
and magnetic phenomena. In the case of a perfectly arbitrary system of forces, there will generally be
interstices between the tubes; but in the case of electric and magnetic forces it is possible to arrange
the tubes so as to leave no interstices. The tubes will then be mere surfaces, directing the motion of
a fluid filling up the whole space. It has been usual to commence the investigation, of the laws of
these forces by at once assuming that the phenomena are due to attractive or repulsive forces acting
between certain points. We may however obtain a different view of the subject, and one more suited
to our more difficult inquiries, by adopting for, the definition of the forces of which we treat, that
they may be represented in magnitude and direction by the uniform motion of an incompressible
fluid.
I propose, then, first to describe a method by which the motion of such a fluid can be clearly
conceived; secondly to trace the consequences of assuming certain conditions of motion, and to point
out the application of the method to some of the less complicated phenomena of electricity, magnetism,
and galvanism; and lastly to skew how, by an extension of these methods, and the introduction of
another idea due to Faraday, the laws of the attractions and inductive actions of magnets, and currents
may be clearly conceived, without making any assumptions as to the physical nature of electricity, or
adding anything to that which has been already proved by experiment.
By referring everything to the purely geometrical idea of the motion of an imaginary fluid, I hope
to attain generality and precision, and to avoid the dangers arising from a premature theory professing
to explain the cause of the phenomena. If the results of mere speculation which I have collected are
found [335 ] to be of any use to experimental philosophers, in arranging and interpreting their results,
they will have served their purpose, and a mature theory, in which physical facts will be physically
explained, will be formed by those who by interrogating Nature herself can obtain the only true
solution of the questions which the mathematical theory suggests.